The major and minor modes are different. Minor scale

Dedicated to L.G. and A.G., muses and fairies, who disenchanted my sense of beauty...

Low music began to play softly. Her unhurried minor chords smoothly flowed around, taking us somewhere into the deep distance. For some reason, there was a whiff of sadness... then the tempo began to increase, high notes were replaced by low ones, the tension gradually increased, and finally a bright, solemnly joyful, major denouement sounded. What happened to us? Mystery of nature...

To avoid ambiguity, I will give a few introductory phrases to clarify the terminology.

As is known, any audio signal of limited duration can be represented as an equivalent Fourier series (spectrum) as a sum of “pure” tones (sinusoidal oscillations) with different amplitudes, frequencies and initial phases. In this work we will consider mainly stationary sound signals that do not change over time.

According to the fundamental tone (first harmonic) of a sound, the lowest frequency of sound is called. All other frequencies above the fundamental tone are called overtones. That. The first overtone is the 2nd highest tone in the sound spectrum. An overtone with a frequency N times greater than the frequency of the fundamental tone (where N is an integer greater than 1) is called the Nth harmonic.

Musical (or harmonic) is a sound that consists only of a set of harmonics. In practice, this is a sound in which all overtones approximately fall within the harmonic frequencies, and some arbitrary harmonics may be absent, including the first. In this case, the fundamental tone is called “virtual” and its pitch will be determined by the psyche of the subject-listener from the frequency ratios between real overtones.

One musical sound may differ from another in fundamental frequency (pitch), spectrum (timbre) and volume. In this work, these differences will not be used, but all our attention will be focused on the mutual relationship of the pitches of sounds.

We will consider the effects of listening to one or more joint musical sounds taken outside of any other musical context.

As is known, the simultaneous sound of two musical sounds of different pitches (two-voice chord, dyad, consonance) can produce in the subject the impression of a pleasant (harmonious, harmonious) or unpleasant (irritating, rough) combination. In music, this impression of consonance is called consonance and dissonance, respectively.

It is also known that the simultaneous sound of three (or more) musical sounds of different pitches (three-voice chord, triad, triad) can produce in the subject an emotional impression of different colors. Different - according to the sign (positive or negative) and strength (depth, brightness, contrast) of the corresponding emotions.

The emotions evoked in people by listening to music, according to their type, among all known emotions, belong to aesthetic (intellectual) and utilitarian emotions. On the classification of emotions, incl. musical see more details.

For example, a triad from the notes “C, E, G” (major) and a triad from the notes “C, E-flat, G” (minor) have, respectively, a pronounced “positive” and “negative” emotional connotation, usually referred to as “joy” and “sadness” (or grief, sadness, suffering, regret, grief, melancholy, despondency - according to).

The emotional coloring of chords is practically independent of changes in the overall pitch, volume or timbre of their constituent sounds. In particular, we will hear an almost unchanged emotional coloring in chords of fairly quiet pure tones.

Looking ahead, we note that if some arbitrary chord can be defined as minor or major, then for the vast majority of subjects the emotions evoked by its sound will be utilitarian, i.e. refers to the category of “sadness or joy” (having a negative or positive sign of emotion). The emotional strength (brightness of emotion) of this chord will generally depend on the specifics of the situation (the state of the subject-listener and the structure of the chord). Essentially (in a statistical sense) it is possible to establish a one-to-one correspondence between major/minor and the emotions they evoke. And most likely it is the emotional coloring of these chords that allows “ordinary people” to recognize the major or minor key of individual chords.

That. Let us summarize that the aesthetic component of the sound “pleasant-unpleasant” (consonance and dissonance) arises in us when listening to two-voice chords, and the emotional component of the sound “joy-sadness” (major and minor) arises in us only when adding a third voice. Note that other types of chords (not major or minor) may not have the utilitarian component of the emotion “contained in them.”

CHORD PROPORTIONS

It is logical to make the assumption that when perceiving a different number of simultaneous musical sounds, the rule of transition of quantity (1, 2, 3 ...) into quality is triggered. Let's see what new qualities can emerge.

Even in ancient times, it was discovered that a chord of two (separately pleasant) sounds can be pleasant or unpleasant (consonant or dissonant) to the ear.

It has been established that such a chord sounds consonant if the ratio of the pitches of its sounds (with an error of, say, 1% or less) is a proportion of relatively small integer (natural) numbers, in particular the numbers from 1 to 6 and 8.

If this proportion consists of relatively large mutually prime numbers (15/16, etc.), then such a chord sounds dissonant.

I note that the accuracy with which whole proportions of musical sounds should be determined, as well as the choice of a specific proportion from a number of alternatives, may depend on the context of the situation. A brief historical excursion into musical intervals is given in.

A list of the ratios of the pitches of two musical sounds (musical intervals) in order of decreasing consonance accordingly looks like this: 1/1, 2/1, 3/2, 4/3, 5/4, 8/5, 6/5, 5/3, and further dissonances 9/5, 9/8, 7/5, 15/8, 16/15.

This list may not be completely complete (at least in terms of dissonances), because is based on possible musical intervals within the framework of an equal-tempered scale of 12 notes per octave (RTS12).

It is also known that the perception of consonance and dissonance occurs at an intermediate level of the human nervous system, at the stage of preliminary processing of individual signals from each ear. If you use headphones to separate two sounds into different ears, then the effects of their “interaction” (consonance peaks, virtual height) disappear.

Digging a little aside, I note that although today there are more than a dozen theories of consonance and dissonance, it is very difficult to give a clear explanation of why the interval 7/5 is dissonance, and 8/5 is consonance (moreover, more perfect than, for example, 5/3) .

However, by and large, we don’t need this here. Is it a good topic for a separate study?

So, let's note the following new fact. When moving from listening to one musical sound to two simultaneous sounds, the subject becomes able to extract information from the pitch ratio of these sounds. Moreover, the psyche of the subject especially distinguishes the ratios of heights in the form of proportions from relatively small natural numbers, which are ranked in one category - consonance/dissonance.

Now let's move on to considering chords of three sounds. In triads, compared to consonances, the number of (pairwise) intervals increases to three, and in addition, a new entity appears - the “monolithic” triad itself (as if a “triple” interval) - the general relationship between the pitches of all three sounds considered together.

This monolithic ratio can be written as the "direct" proportion A:B:C or in another form as the "inverse" proportion (1/D):(1/E):(1/F) of natural coprime triplets of numbers A, B,C or D,E,F. Purely mathematically, all such proportions can be divided into three main groups:

The direct proportion is “simpler” than the inverse, i.e. A*B*C< D*E*F

The inverse proportion is “simpler” than the direct one, i.e. A*B*C > D*E*F

Both proportions are the same (“symmetrical”), i.e. A*B*C = D*E*F (and so A=D, B=E, C=F).

That. a new quality of triad - information of a new type - can only be contained in these triple proportions, falling into one of the three categories described above.

Depending on the degree of consonance of all paired intervals, triads can be either consonant or dissonant. In some cases (when using various integer approximations), the choice of the specific composition of both proportions may be ambiguous. However, for consonantal chords such ambiguity does not appear.

According to musical practice, there are four main types of triads - major and minor (consonances), augmented and diminished (dissonances). Almost all consonant chords can be classified as major or minor.

The pitch ratios of the above-mentioned major triad with great accuracy are in direct proportion 4:5:6. The pitch ratios of the above minor triad with great accuracy are the inverse proportion /6:/5:/4. The direct and inverse proportions of the augmented and diminished triads are the same, because they consist of equal intervals (4-4 and 3-3 semitones RTS12), and these equal proportions look like /25:/20:/16 = 16:20:25 and, accordingly, /36:/30:/25 = 25: 30:36.

The pitch relationships of major triads are always more simply (using smaller integers) expressed in direct proportions, and of minor triads in inverse proportions, and this is a well-known fact. Already Gioseffo Zarlino (1517-1590) knew the opposite meaning of major and minor chords (“Istituzione harmoniche” 1558). However, even 450 years later, it is not so easy to find a serious work in which this fact is widely used for harmonic analysis or chord synthesis. The reason for this may have been the persistent but erroneous attempts of various authors to explain the phenomenon of major and minor (see below). Maybe the connection between chords and pitch proportions has become something of a forbidden topic of “perpetual motion”?

Based on simple mathematics and experimental data, we put forward a postulate: any major chord (it is simpler in direct proportion) can be turned into a minor chord (it is simpler in inverse proportion), if instead of a direct proportion we write the inverse of the same numbers. Those. if the proportion A:B:C is major, then the inverse (different!) proportion /C:/B:/A is minor. Of course, any direct proportion can (without changes!) be represented as an inverse, and vice versa. In particular, 4:5:6 = /15:/12:/10 and /4:/5:/6 = 15:12:10.

Summarizing all this, we can conclude that the three groups into which all proportions of triad pitches are divided really play an important role in musical practice, and correspond to the division of chords into major, minor and “symmetrical” (consisting of identical intervals).

One might ask: what is the “internal” representation of musical triads in the psyche of the subject? How does he use information about the above-mentioned “new quality” of the triad?

Taking into account the highly developed apparatus of the human auditory system, it can be assumed that although the human higher nervous system is quite capable of representing a minor triad in the form of a direct proportion (15:12:10), it is also (if not easier) capable of representing the same triad in in the form of an inverse proportion (/4:/5:/6), and “at the first comparison” of these proportions (to determine the category), “discard” the straight line because of its 15 times greater complexity (the product of three numbers of direct and inverse proportions is equal to 1800 versus 120).

We will further call the main proportion of a chord one of the two proportions of the pitches of its sounds (direct or inverse), which consists of smaller numbers (in the sense of their product), while the other proportion will be called secondary. That. The main proportion of a major chord will always be a direct proportion, and a minor chord will always be an inverse proportion.

And finally, we note that although the above-mentioned minor and major triads consist in pairs of the same intervals (4:5, 4:6, 5:6), they have the opposite emotional connotation, which is absent in any individual pair of their sounds. The only difference between monolithic triads (minor and major) is the fact of mutual reversal of their main proportions.

It is logical to conclude that the corresponding new “emotional” information of the chord is contained precisely in this last property (type of principal proportion), which can only appear when three or more sounds are combined, but cannot be detected when two are combined (since, say A:B is absolutely the same as /A:/B). There is simply no other source of (emotional) information contained in the triad (don’t forget that we are considering stationary sounds with a constant spectrum). Additional confirmation of this conclusion is that the sound of “symmetrical” chords does not have a utilitarian component of emotions.

Example 1. Sound of proportions

2:3:4 = /6:/4:/3 gives a soft major. 2:3:6 = /3:/2:/1 gives a soft minor.

3:4:5 = /20:/15:/12 gives a brighter (contrasting) major, and 20:15:12 = /3:/4:/5 gives a deeper (contrasting) minor.

4:5:6 = /15:/12:/10 gives the brightest major, and 10:12:15 = /6:/5:/4 gives the deepest minor.

To listen to chords, it is better to use pure tones with precise frequency relationships, using e.g. .

THEORIES OF MAJOR AND MINOR

Chords have been heard in music for many hundreds of years, and people have been thinking about the reasons for their euphony for almost as long.

For two-voice chords, the first explanation of this property was made a very long time ago (and it is captivatingly simple and clear, if you close your eyes to some dissonances - see above). For three-voice chords of major and minor, the above-described facts about direct and inverse proportions were also established quite a long time ago.

However, finding an answer to the question of why different chords have different emotional colors in sign (and strength) turned out to be much more difficult. And to the second question - why does a minor chord, with all its complexity (when presented in direct proportions - so to speak, in “major notation”) sound euphonious, and let’s say “almost the same” in terms of complexity of the numerical proportion “dischord” (like 9:11 :14) sounds unpleasant - it was difficult to answer.

Generally speaking, it was not entirely clear how to justify “equally well” both major and minor?

Many authoritative researchers have tried to figure out this mystery of the nature of major and minor. And if the major was explained “quite simply” (as it seemed to many authors, for example, “purely acoustically”), then the problem of justification for the minor, which is similar in clarity, apparently is still on the agenda, although there are a great many different theoretical and phenomenological constructions trying to provide its solution.

The interested reader may refer to .

History- but it does not have to always be the case - for example in the case of a chord of pure tones.

Some authors, when “justifying” chords, also referred to the nonlinear properties of hearing, described for example. V . However, this undeniably true fact very rarely works in practice, because even a chord that is not too weak in volume will not generate distinguishable combination tones due to nonlinearity.

Other authors used very complex theoretical and musical constructions (or purely mathematical schemes, closed as “things in themselves”), the exact meaning of which was often impossible to understand without a detailed study of the specific terminology of these theories themselves (and sometimes this explanation was based on paraphrasing some abstract terms through others).

Some authors are still trying to approach this issue from the point of view of cognitive psychology, neurodynamics, linguistics, etc. And they almost succeed... Almost - because the chain of explanations can be too long and far from indisputable, and besides, there is no algorithmic formalization of theories, etc. basis for their quantitative experimental verification.

For example, in one of the most interesting, detailed and versatile studies of the phenomenon of major and minor, a hypothesis is given that the basis for the emotional content of sounds was laid down by nature in the instinct of higher animals, which was further developed in humans. It has been experimentally established that the dominance of a particular individual of a pack in the animal world is accompanied by the use of low or descending “speech” sounds, and subordination is accompanied by the use of high or rising sounds. It is further accepted that dominance is equal to “joy”, and submission is equal to “sadness”. Then a table is constructed of dissonant symmetrical triadic chords (with two equal intervals from 1 to 12 semitones RTS12) with a list of changes in these chords to minor when increasing or to major when the pitch of any sound of the original chord is decreased by one semitone.

Even aside from the fact that some of the changed chords cannot be unambiguously classified as major or minor, it is not clear why, when listening to a chord, a human subject must necessarily (and instantly) “think” that one of the sounds of this (consonantal) chord is shifted from the sound of another (unambiguously defined and also dissonant) chord for a certain fixed interval - a semitone? And how can this rather abstract thought turn into “innate” emotions? And why should the mind be limited only to the capabilities of the RTS12? RTS12 was also invented by Nature and put into instinct?

However, I agree that the emotional content of major and minor is based on the emotions available to many higher animals... although it is unclear - can they experience these emotions when listening to chords? I think it's unlikely. Because determining the relative proportions of the pitches of three or more sounds of a chord is a process of a higher order of complexity than determining the pitch of one sound (or the direction of change in this pitch).

The human hearing system has received particular development in connection with the advent of speech communication, which has given rise to the ability to perform detailed and rapid analysis of the spectrum of complex sounds, a by-product of which is most likely our ability to enjoy music.

Utilitarian emotions in higher animals (as well as in humans), however, may well be evoked through the perception of information from other senses - and above all - through the visual perception of events and their further interpretation.

A few words about the emotionality of human speech and monophonic music. Yes, they may “contain” utilitarian emotions. But the reason for this is the significant non-stationarity of the spectrum - changes in the height and/or timbre of these sounds.

And also about the individual differences of the subjects. Yes, with the help of special education (training) it is possible to accustom people (as well as some animals) to the fact that even one sound (or any chord) will evoke utilitarian emotions in them (grief from a reflexively expected stick or joy from a carrot ). But this will not correspond to the natural nature of things which we seek to establish.

Here is a phrase from a doctoral dissertation in musicology in 2008, apparently putting an end to the question of the well-known theories of major and minor: “despite the fact that many authors have described the perception of major/minor chords and scales, it still remains a mystery why major chords feel happy and minor chords feel sad.”

I think that the development of a correct theory of major and minor is possible only if two important conditions are satisfied:

Involving additional areas of knowledge (except music and acoustics), - using the mathematical apparatus of additional areas of knowledge.

We should remember history. The idea that the “meaning” of a chord should be sought outside the “old” space of music theory was first heard at least more than a hundred years ago.

Here are a couple of quotes.

Hugo Riemann (1849-1919) towards the end of his career abandoned the justification of major and consonance through the phenomenon of overtones and adopted the psychological point of view of Karl

Stumpf, considering overtones only as “an example and confirmation”, but not proof.

Karl Stumpf (1848-1936) transferred the scientific basis of music theory from the field of physiology to the field of psychology. Stumpf refused to explain consonance as an acoustic phenomenon, but proceeded from the psychological fact of “fusion of tones” (Stumpf C.Tonpsychologie. 1883-1890).

So, concluding this section, I note that most likely Stumpf and Riemann were absolutely right that it is impossible to substantiate a chord either acoustically, metaphysically, or purely musically, and what is necessary for this is the involvement of psychology.

Now let’s approach the question “from the other end” and ask the question: what is emotion?

THEORIES OF EMOTIONS

Let us briefly consider two theories of emotions, which, in my opinion, come closest to the level at which the possibility of applying their laws in such a complex issue as the psychological structure of the phenomena of music perception opens up.

For other theories and details, I refer the reader to a rather extensive review in.

Frustration theory of emotions

In the 1960s L. Festinger's theory of cognitive dissonance arose and was thoroughly developed.

According to this theory, when there is a discrepancy between expected and actual performance results (cognitive dissonance), negative emotions arise, while the coincidence of expectations and results (cognitive consonance) leads to positive emotions. The emotions that arise during dissonance and consonance are considered in this theory as the main motives for the corresponding human behavior.

Despite many studies confirming the correctness of this theory, there is also other evidence showing that in some cases, cognitive dissonance can cause positive emotions.

According to J. Hunt, for the emergence of positive emotions, a certain degree of discrepancy between attitudes and signals, a certain “optimum of discrepancy” (novelty, unusualness, inconsistency, etc.) is necessary. If the signal does not differ from previous ones, then it is assessed as uninteresting; if it differs too much, it seems dangerous, unpleasant, annoying, etc.

Information theory of emotions

Somewhat later, an original hypothesis about the causes of the phenomenon of emotions was put forward by P.V. Simonov.

According to it, emotions appear as a result of a lack or excess of information necessary to satisfy the subject’s needs. The degree of emotional stress is determined by the strength of the need and the magnitude of the deficit of pragmatic information necessary to achieve the goal.

P.V. Simonov considered the advantage of his theory and the “formula of emotions” based on it to be that it contradicts the view of positive emotions as a satisfied need. From his point of view, a positive emotion will arise only if the received information exceeds the previously existing forecast regarding the probability of satisfying the need.

Simonov’s theory was further developed in the works of O.V. Leontiev, in particular, by 2008, a very interesting article was published with a number of generalized formulas of emotions, one of which I will describe in detail below. I quote further.

By emotions we mean a mental mechanism for controlling the behavior of a subject, assessing the situation according to a certain set of parameters... and launching the corresponding program of his behavior. In addition, each emotion has a specific subjective coloring.

The above definition assumes that the type of emotion is determined by the corresponding set of parameters. Two different emotions must differ in a different set of parameters or range of their values.

In addition, psychology describes various characteristics of emotions: sign and strength, time of occurrence relative to the situation - preceding (before the situation) or stating (after the situation), etc. Any theory of emotions must make it possible to objectively determine these characteristics.

The dependence of an emotion on its objective parameters is called the formula of emotions.

One-parameter formula of emotions

If a person has a certain need of value P, and if he manages to obtain a certain resource Ud (for Ud > 0) that satisfies the need, then the emotion E will be positive (and in the case of loss of Ud< 0 и эмоция будет отрицательной):

E = F(P, UD) (1)

The resource Ud is defined in the work as the “Level of Achievement”, and the emotion E is defined as a stating one.

To be specific, you can imagine a person playing a new game and having no idea what to expect from it.

Joy.

If a player wins a certain amount Ld > 0, then a positive emotion of joy arises with force

E = F(P, Ud).

Grief.

If the player “won” the amount< 0 (т.е. проиграл), то возникает отрицательная эмоция горя

force E = F(P, Ud).

Another method of formalizing emotions is proposed in the work.

According to him, emotions are considered as a means of optimal control of behavior, directing the subject to achieve the maximum of his “target function” L.

An increase in the target function L is accompanied by positive emotions, a decrease - by negative emotions.

Since L depends in the simplest case on some variable x, emotions E are caused by a change in this variable over time:

E = dL/dt = (dL/dх)*(dх/dt) (2)

It is also noted that, along with the above-described (utilitarian) emotions, there are also so-called. “intellectual” emotions (surprise, guess, doubt, confidence, etc.), which arise not in connection with a need or goal, but in connection with the intellectual process of information processing itself. For example, they can accompany the process of observing abstract mathematical objects. A feature of intellectual emotions is their lack of a specific sign.

At this stage, we will stop quoting and move mainly to the presentation of the author’s original ideas.

MODIFICATION OF FORMULAS OF EMOTIONS

First of all, we note that formulas (1, 2) are very similar, if we take into account that the resource parameter Ud is actually the difference between the current and previous value of a certain integral resource R. For example, in the case of our gambler, it is logical to choose his total capital as R , Then:

BP = R1 - R0 = dR = dL

However, both formulas (1, 2) are “not entirely” physical - they equate quantities that have different dimensions. You can’t measure, say, time in kilometers or joy in liters.

Therefore, firstly, the formulas of emotions should be modified by writing them in relative quantities.

It is also desirable to clarify the dependence of the strength of emotions on their parameters, i.e. to increase the plausibility of the results over a wide range of changes in these parameters.

To do this, we will use an analogy with the well-known Weber-Fechner law, which says that the differential threshold of perception for a variety of human sensory systems is proportional to the intensity of the corresponding stimulus, and the magnitude of the sensation is proportional to its logarithm.

In fact, the joy of that same player should be proportional to the relative size of the win, and not to the absolute size. After all, a billionaire who lost one million will not grieve as much as the owner of a million or so. And the heights of the “most similar” musical sounds are related by octave ratio, i.e. also logarithmic (increasing the frequency of the fundamental tone of the sound by 2 times).

I propose to write the modified emotion formula (1) as follows:

E = F(P) * k * log(R1/R0), (3)

where F(P) is a separate dependence of emotions on the need parameter P;

k is some constant (or almost constant) positive value, depending on the subject area of ​​the resource R, on the base of the logarithm, on the time interval between measurements R1 and R0, and also possibly on the details of the character of a particular subject;

R1 is the value of the target function (total useful resource) at the current point in time, R0 is the value of the target function at the previous point in time.

You can also express the new formula of emotions (3) through the dimensionless quantity L = R1/R0, which is logically called the relative differential objective function (the current value of the integral objective function relative to some previous moment in time, always located at a fixed distance from the current moment).

E = F(P) * Pwe, where Pwe = k * log(L), (4)

where in turn L = R1/R0, and the parameters k, R0 and R1 are described in formula (3).

Here the value of the power of emotions Pwe is introduced, proportional to the “flow of emotional energy” per unit of time (i.e., the everyday meaning of the expression “intensity of emotions”, “strength of emotions”). The expression of the strength of emotions in units of power allocated by the subject’s body to emotional behavior is known from the works of other authors, so we should not be surprised at the appearance of such a (somewhat unusual) term as “power of emotion.”

As is easy to see, formulas (3 and 4) automatically give the correct sign of emotions, positive when R rises (when R1 > R0 and thus L > 1) and negative when R falls (when R1< R0 и т.о. L < 1).

Now let's try to apply new formulas of emotions to the perception of musical chords.

INFORMATION THEORY OF CHORDS

But first, a little “lyrics”. How can the above information theory of emotions be expressed in simple human language? I’ll try to give a few fairly simple examples to clarify the situation.

Let’s say that today life has given us a “double portion” of some “life benefits” (against the average daily amount of “happiness”). For example - twice the best lunch. Or we had two hours of free time in the evening versus one. Or we walked twice as far as usual on a mountain hike. Or we were given twice as many compliments as yesterday. Or we received double bonuses. And we are happy because the function L today has become equal to 2 (L=2/1, E>0). And tomorrow we will receive all this fivefold. And we rejoice even more (we experience more powerful positive emotions, because L=5/1, E>>0). And then it all went as usual (L=1/1, E=0), and we no longer experience any utilitarian emotions - we have nothing to be happy about, and nothing to be sad about (if we have not yet gotten used to happy days). And then suddenly a crisis broke out and our benefits were cut in half (L = 1/2, E<0) - и нам стало грустно.

And although for each subject the goal function L depends on a large set of individual sub-goals (sometimes diametrically opposed - for sports opponents or fans, for example), what is common to all is the personal opinion of each - whether this event brings him closer to some of his goals, or moves away from them.

Now let's get back to our music.

Based on verified facts of science, it is logical to assume that when listening to several sounds simultaneously, the subject’s psyche tries to extract all kinds of information that these sounds may contain, including those located at the highest level of the hierarchy, i.e. from the pitch ratios of all sounds.

At the stage of analyzing the parameters of triads (as opposed to consonances, see above), individual streams of information from different ears are already used together (which is easy to check by sending any two sounds to one ear, and the third to the other - the emotions are the same).

In the process of interpreting this combined information, the subject’s psyche tries to use, among other things, its “utilitarian” emotional subsystem.

And in some cases she succeeds in this - for example, when listening to isolated minor and major chords (but other types of chords can apparently generate other types of emotions - aesthetic / intellectual).

Perhaps some fairly simple analogies (at the level of more/less) with the meaning of “similar” information from other sensory channels of perception (visual, etc.) allow the subject’s psyche to classify major chords as carrying information “about benefit”, accompanied by positive emotions, and minor chords - “about loss”, accompanied by negative ones.

Those. in the language of the emotion formula (4), the major chord should contain information about the value of the objective function L > 1, and the minor chord should contain information about the value of L< 1.

My main hypothesis is the following. When perceiving a separate musical chord, the value of the target function L is generated in the subject’s psyche, which is directly related to the main proportion of the pitches of its sounds. In this case, major chords correspond to the idea of ​​an increase in the target function (L>1), accompanied by positive utilitarian emotions, and minor chords correspond to the idea of ​​a decline in the target function (L<1), сопровождаемое отрицательными утилитарными эмоциями.

As a first approximation, we can assume that the value of L is equal to some simple function of the numbers included in the main proportion of the chord. In the simplest case, this function can be some kind of “average” of all the numbers of the main proportion of the chord, for example, the geometric mean.

For any major chords, all these numbers will be greater than 1, and for any minor chords they will be less than 1.

For example:

L = N = "average" of the numbers (4, 5, 6) from the major proportion 4: 5: 6,

L = 1/N = “average” of the numbers (1/4, 1/5, 1/6) from the minor proportion /4:/5:/6.

With this representation of L, the amplitude of the strength of emotions (i.e., the absolute value of Pwe) generated by the major and (its inverse) minor triad will be exactly the same, and these emotions will have the opposite sign (major - positive, minor - negative). A very encouraging result!

Let us now try to clarify and generalize formula (4) for an arbitrary number of voices of the chord M. To do this, we define L as the geometric mean of the numbers from the main proportion of the chord, ultimately obtaining the final form of the “formula of musical emotions”:

Pwe = k * log(L) = k * (1/M) * log(n1 * n2 * n3 * ... * nM), (5)

where k is still some positive constant - see (3),

Let's call the value Pwe (from formula 5) the “emotional power” of the chord (or simply power), positive for a major and negative for a minor (analogy: the flow of vital forces, for a major there is an influx, for a minor there is an outflow).

For consistency with the logarithmic frequency scale (remember the octave), we will use the base 2 logarithm in formula (5). In this case, we can set k = 1, because in this case, the numerical value of Pwe will be in a completely acceptable range near the region of “unit” amplitude of emotions.

For further analysis, along with the “main” one, we may also need the “side” power of the chord, corresponding to the substitution of its side proportion into formula (5) (see above). If not specified, the “main” Pwe is used throughout below.

The appendix to the article shows the meanings of the main and secondary powers of some chords.

THE DISCUSSION OF THE RESULTS

So, having put forward a number of fairly simple and logical assumptions, we have obtained new formulas (3, 4, 5), which connect the general parameters of the situation (or specific parameters of the chords for formula 5) with the sign and strength of the utilitarian emotions they evoke (in the context of the situation).

How can we evaluate this result?

I quote the work:

“There have probably been no attempts to objectively determine the strength of an emotion. However, it can be assumed that such a definition should be based on energy concepts. If an emotion causes some behavior, then this behavior requires a certain expenditure of energy. The stronger the emotion, the more intense the behavior, the more energy is required per unit of time.

Those. We can try to identify the strength of an emotion with the amount of power that the body allocates to the corresponding behavior.”

Let's try to approach the new result as critically as possible, since there is nothing to compare it with yet.

Firstly, the power of emotions Pwe from formulas (4, 5) is proportional to the “subjective strength” of emotions, but their relationship may not be linear. And this connection is only a certain average dependence along the entire continuum of subjects, i.e. may be subject to significant (?) individual variations. For example, the “constant” k can still change, although not too much. It is also possible that instead of the geometric mean in formula (5), some other function should be used.

Secondly, if we keep in mind the specific form of the formula for musical emotions (5), then it should be noted that although formally in it M can be equal to 1 or 2, we can talk about the emergence of utilitarian emotions only when M >= 3. However, already with M = 2, the presence of aesthetic/intellectual emotions is possible, and with M > 3, there is a possibility of additional factors (?) somehow influencing the result.

Thirdly, apparently the range of valid values ​​of the Pwe amplitude for the categories of major and minor has an upper limit of 2.7 ... 3.0, but somewhere already from the value of 2.4 the area of ​​saturation of the utilitarian-emotional perception of chords begins, and the lower limit of the range passes approximately there possible “invasion” of dissonances.

But this latter is rather a general problem of the “non-monotonicity” of a number of dissonant intervals, not directly related to the emotional perception of chords. And the limited dynamic range of the power of emotions is a general property of any human sensory system, easily explained by the lack of analogies with events in “real life”, which correspond to too rapid changes in the target function (7-8 times or more).

Fourthly, “symmetrical” (or almost symmetrical) chords, in which direct and inverse proportions consist of the same numbers (even in the absence of obvious dissonances in them) apparently fall out of our classification - their utilitarian-emotional coloring is practically absent, corresponding to the case Pwe = 0.

However, it is possible to supplement the formal result of applying formula (5) with a simple semi-empirical rule: if the main and secondary powers of some chord (almost) coincide in amplitude, then the result of formula (5) will not be the main power, but half the sum of the powers, i.e. (approximately) 0.

And this rule begins to work already when the difference between the amplitudes of the main and secondary Pwe is less than 0.50.

Most likely, a very simple phenomenon is taking place here: since it is impossible to distinguish the direct and inverse proportions of a chord by complexity, then the classification of this chord in the categories of utilitarian emotions (“sadness and joy”) is simply not carried out. However, these chords (like intervals) can generate aesthetic/intellectual emotions, e.g. “surprise”, “question”, “irritation” (in the presence of dissonances), etc.

With all its imaginary or real shortcomings, formula (5) (and, apparently, formulas 3 and 4) still gives us very good theoretical material for numerical estimates of the strength of emotions.

At least in one specific area - the area of ​​\u200b\u200bthe emotional perception of major and minor chords.

Let's try to test this formula (5) in practice, by comparing a pair of different major and minor chords. A very good example is the chords 3:4:5 and 4:5:6 and their minor variations.

For the purity of the experiment, pairs of chords composed of pure tones should be compared at approximately the same average volume level, and for both chords it is better to use such pitches that the “weighted average” frequency of these chords (in Hertz) is the same.

A pair of major triads can consist of tones of frequency e.g. 300, 400, 500 Hz and 320, 400, 480 Hz.

From the ear, it seems quite noticeable to me that the emotional “brightness” of the major 3:4:5 (with Pwe = 1.97) is indeed somewhat less than that of the major 4:5:6 (with Pwe = 2.30). In my opinion, approximately the same thing happens with the minor /3:/4:/5 and /4:/5:/6.

This impression of correct transmission of the power of emotions by formula (5) is also preserved when listening to the same chords composed of sounds with a rich harmonic spectrum.

TOTAL

In total, in accordance with the information theory of emotions, the work proposes modified formulas that express the sign and amplitude of utilitarian emotions through the parameters of the situation.

A hypothesis has been put forward that when a musical chord is perceived, the value of a certain target function L is generated in the subject’s psyche, which is directly related to the proportion of the pitches of the chord sounds. In this case, major chords correspond to direct proportions, giving rise to the idea of ​​an increase in the target function (L>1), causing positive utilitarian emotions, and minor chords correspond to inverse proportions, giving rise to the idea of ​​a decline in the target function (L<1), вызывающее отрицательные утилитарные эмоции.

A formula for musical emotions has been put forward: Pwe = log(L) = (1/M)*log(n1*n2*n3* ... *nM), where M is the number of voices of the chord, ni is an integer (or reciprocal fraction) from the general proportion of pitches corresponding to the i-th voice of the chord.

A limited experimental test was carried out, the limits of applicability of the formula of musical emotions were explored, in which it correctly conveys the sign and (in my opinion) their amplitude.

CODA

The fanfare sounds joyfully!

Then everyone stands up - and holding hands - a cappella they sing the Hymn to Reason!

The centuries-old mystery of major and minor has finally been solved! We won...

LITERATURE AND LINKS

1. Audiere sound system, Download archive Use wxPlayer.exe from the bin folder.
2. Trusov V.N. Materials from the site mushar.ru 2004 http://web.archive.org/http://mushar.ru/
3. Mazel L. Functional school. 1934 (Ryzhkin I., Mazel L., Essays on the history of theoretical musicology)
4. Riman G. Musical dictionary (computer version). 2004
5. Leontyev V.O. Ten unresolved problems in the theory of consciousness and emotions. 2008
6. Ilyin E.P. Emotions and feelings. 2001
7. Simonov P.V. Emotional brain. 1981
8. Leontyev V.O. Formulas of emotions. 2008
9. Aldoshina I., Pritts R. Musical acoustics. 2006
10. Aldoshina I. Fundamentals of psychoacoustics. A selection of articles from the site http://www.625-net.ru
11. Morozov V.P. The art and science of communication. 1998
12. Altman Ya.A. (ed.) The auditory system. 1990
13. Lefebvre V.A. Formula of man. 1991
14. Shiffman H.R. Sensation and perception. 2003
15. Teplov B.M. Psychology of musical abilities. 2003
16. Kholopov Yu.N. Harmony. Theoretical course. 2003
17. Golitsyn G.A., Petrov V.M. Information - behavior - creativity. 1991
18. Garbuzov N.A. (ed.) Musical Acoustics. 1954
19. Rimsky-Korsakov N. Practical textbook of harmony. 1937
20. Leontyev V.O. What is an emotion? 2004
21. Klaus R. Scherer, 2005. What are emotions? And how can they be measured? Social Science Information, Vol 44, no 4, pp. 695-729
22. BEHAVIORAL AND BRAIN SCIENCES (2008) 31, 559-621 Emotional responses to music: The need to consider underlying mechanisms
23. Music Cognition at the Ohio State University http://csml.som.ohio-state.edu/home.html Music and Emotion http://dactyl.som.ohio-state.edu/Music839E/index.html
24. Norman D. Cook, Kansai University, 2002. Tone of Voice and Mind: The connections between intonation, emotion, cognition and consciousness.
25. Bjorn Vickhoff. A Perspective Theory of Music Perception and Emotion. Doctoral dissertation in musicology at the Department of Culture, Aesthetics and Media, University of Gothenburg, Sweden, 2008
26. Terhardt E. Pitch, consonance, and harmony. Journal of the Acoustical Society of America, 1974, Vol. 55, pp. 1061-1069.
27. VOLODIN A.A. Abstract of doctoral dissertation. PSYCHOLOGICAL ASPECTS OF PERCEPTION OF MUSICAL SOUND
28. Levelt W., Plomp R. The appreciation of musical intervals. 1964

ACKNOWLEDGMENTS

I express my gratitude to Ernst Terhardt and Yury Savitski for the literature kindly provided to me for writing this work. Thank you very much!

AUTHOR'S INFORMATION

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Version.

APPLICATION

Emotional power Pwe of the main proportions of some chords, calculated using formula (5).

The main part of the proportions are direct proportions corresponding to major chords.

Minor chords can be made from proportions that are the reverse of major chords by simply changing the Pwe sign of the major proportion (as in a couple of examples).

The side power of some chords is given in parentheses if it approaches the main power in amplitude.

For symmetrical chords, both of these powers differ only in sign.

Home Side Pwe main (side) Note proportion proportion proportion

Some symmetrical [pseudo]chords

1:1:1 1:1:1 0 (0)

1:2:4 /4:/2:1 1 (-1)

4:6:9 /9:/6:/4 2.58 (-2.58) “fifth” triad

16:20:25 /25:/20:/16 4.32 (-4.32) increased triad

1:2:3 /6:/3:/2 0.86 (-1.72)

2:3:4 /6:/4:/3 1.53 (-2.06)

2:3:5 /15:/10:/6 1.64

2:3:8 /12:/8:/3 1.86

2:4:5 /10:/5:/4 1.77

2:5:6 /15:/6:/5 1.97

2:5:8 /20:/8:/5 2.11

3:4:5 /20:/15:/12 1.97 /3:/4:/5 20:15:12 -1.97

3:4:6 /4:/3:/2 -1.53 (2.06)

3:4:8 /8:/6:/3 2.19 (-2.39) almost symmetrical

3:5:6 /10:/6:/5 2.16 (-2.74)

3:5:8 /40:/24:/15 2.30

3:6:8 /8:/4:/3 2.39 (-2.19) almost symmetrical

4:5:6 /15:/12:/10 2.30 major triad

/4:/5:/6 15:12:10 -2.30 minor triad

4:5:8 /10:/8:/5 2.44 (-2.88)

5:6:8 /24:/20:/15 2.64

Some dissonant triads

4:5:7 /35:/28:/20 2.38

5:6:7 /42:/35:/30 2.57

1:2:3:4 /12:/6:/4:/3 1.15

2:3:4:5 /30:/20:/15:/12 1.73

3:4:5:6 /20:/15:/12:/10 2.12

You already know that most often music is recorded in major and minor modes. Both of these modes have three varieties - natural scale, harmonic scale and melodic scale. There is nothing terrible behind these names: the basis is the same for all, only in harmonic and melodic major or minor certain steps (VI and VII) change. In a minor they will go up, and in a major they will go down.

3 types of major: first – natural

Natural major- this is an ordinary major scale with its key signs, if they exist, of course, and without any random alteration signs. Of the three types of major, this one is found more often than others in musical works.

The major scale is based on the well-known formula of the sequence in the scale of whole tones and semitones: T-T-PT-T-T-T-PT . You can read more about this.

Look at examples of several simple major scales in their natural form: natural C major, the G major scale in its natural form, and the scale of the key of natural F major:

3 types of major: the second is harmonic

Harmonic major– this is a major with a lower sixth degree (VIb). This sixth step is lowered in order to be closer to the fifth. The low sixth degree in major sounds very interesting - it seems to “minorize” it, and becomes gentle, acquiring shades of oriental languor.

This is what the harmonic major scales of the previously shown keys C major, G major and F major look like.

In C major, A-flat appeared - a sign of a change in the natural sixth degree, which became harmonic. In G major the sign E-flat appeared, and in F major - D-flat.

3 types of major: third – melodic

As in , in the major of that variety, two steps change at once - VI and VII, only everything here is exactly the opposite. Firstly, these two sounds do not rise, as in minor, but fall. Secondly, they alter not during an upward movement, but during a downward movement. However, everything is logical: in the melodic minor scale they rise in an ascending movement, and in the melodic minor scale they decrease in a descending movement. It seems like this is how it should be.

It is curious that due to the lowering of the sixth stage, all sorts of interesting intervals can form between this stage and other sounds - increased and decreased. It could be or - I recommend that you look into it.

Melodic major- this is a major scale in which, with an upward movement, a natural scale is played, and with a downward movement, two steps are lowered - the sixth and seventh (VIb and VIIb).

Notation examples of the melodic form - the keys C major, G major and F major:

In melodic C major, two “accidental” flats appear in a descending movement - B-flat and A-flat. In G major of the melodic form, the F-sharp is first canceled (the seventh degree is lowered), and then a flat appears before the note E (the sixth degree is lowered). In melodic F major, two flats appear: E-flat and D-flat.

And one more time...

So, there are . This natural(simple), harmonic(with a reduced sixth stage) and melodic(in which when moving upward you need to play/sing the natural scale, and when moving down you need to lower the seventh and sixth degrees).

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Allegro S. Prokofiev. Classical Symphony, Gavotte

All other degrees of the mode (II, IV, VI and VII) are unstable, and the degree of their instability is determined by two factors: the interval relationship with the nearest stable sounds and the “force of attraction” of a given stable sound. Thus, with a half-tone ratio of the discontinuity to the abutment, the desire for resolution will be felt stronger, more acutely than with their whole-tone ratio. In addition, the sound of the tonic (that is, stage I) “attracts” unstable sounds to itself more strongly than other foundations.

The desire of unstable steps to transform into stable ones is called gravity, and the transition itself to a stable sound - resolution this gravity.

The properties of stability and instability are not inherent in one or another specific sounds of a musical system, namely certain scale degrees. In turn, any mode can be built from each of the twelve sounds of the chromatic series contained in the octave. Thus, the same fret can occur at any height, depending, ultimately, on the location of its first degree (tonic).

Musical sounds themselves are only elements from which, with a certain organization, one or another modal system can be formed. Between the sounds that form any modal system, not only pitch (interval) always arise, but also the so-called modal-functional relationships: each sound, becoming a certain level of the mode, certainly acquires the property of stability or instability and, in accordance with this meaning, fulfills one or another role - function- in mode, which determines the very name of this level (in addition to its musical name). The number of steps of any mode is always strictly defined: they receive serial numbers assigned to them, designated by Roman numerals, which, if necessary, are written below under the corresponding notes.

The same sound in different mode systems can have different step (functional) meaning. However, the very role of functionally identical scale degrees, no matter what sounds they are represented in a particular system, always remains unchanged.

For example, in the melodies of the two Russian folk songs given below, there are many common features: they are close to each other in genre (both are of a dance, round dance nature), structure, size (meter), rhythmic pattern, intonation structure and even sound composition (in all six of them* [In the melodies of both songs, only six sounds of the seven-step scale are used: in the first there is no E sound (that is, the VII step), and in the second there is no E-flat sound (that is, the VI step).] sounds are common), but these songs are written in different keys and modes: the first is in F major, and the second is in G minor.

83 Fun, lively Russian folk song “Dunya held the transport”

Not very soon Russian folk song “I am sitting on a pebble”

The melodies of these songs are specially signed one below the other in such a way that they can be easily compared (in those cases when the same sounds occur in these melodies on the same metrical beats of the corresponding measures, they are connected by a dotted line vertically). In this case, it turns out that the step (functional) meaning of identical sounds is different.

Any modal system can be expressed with sufficient clarity and completeness, both monophonically and polyphonically. However, in polyphony, thanks to the simultaneous sounding of several voices, certain harmonic complexes are formed (individual chords, consonances and entire harmonic turns), which can, on the one hand, contribute to a more vivid and characteristic manifestation of modal functions, and on the other hand - under certain conditions - in able to change the role of a particular sound in a given context.

So, for example, the sound of the V degree as part of a tonic triad is stable, but as part of a dominant harmony (in particular, a dominant seventh chord) the same sound is functionally unstable. Moreover, even the sound of the first degree of a mode, which is certainly stable in a monophonic mode or as part of a tonic triad, may turn out to be unstable with certain harmonic accompaniment. For example:

87 C major

In the above harmonic revolution, the second sound before in the upper voice it comes into conflict with the dominant harmony that arose on the third beat in the other voices and clearly tends to transition into sound si, thereby becoming like an unstable step. However, sound stability before is restored with the next chord change and the transition of an ascending tense leading tone si into tonic (for more details, see § 40).

In the examples given below from literary musical literature, only one melodic line is first given, and then the same line is given, but with the author’s harmonic accompaniment. It is easy to notice how much brighter and more fully the same themes are perceived in the second case, although both melodies themselves are quite bright and characteristic (especially by S. S. Prokofiev):

88 Moderato con moto N. R.-Korsakov. “I’m still full, oh my dear friend...”

89 Vivace S. Prokofiev. "Romeo and Juliet", No. 10

90 Moderato con moto N. R.-Korsakov. “I’m still full, oh my dear friend...”

91 Vivace S. Prokofiev “Romeo and Juliet”, No. 10

In the music of the vast majority of countries in the world, there are two main modes - major and minor. All other mode formations ultimately come down, as a rule, to one or another modification of the major or minor modes. Sometimes found in the professional musical work of composers, as well as in folk music of various countries of the world (such as, for example, Turkey, India and some others), other mode systems are only separate, although interesting, but still private (and sometimes completely exceptional) cases that do not have universal significance.

Major mode(or simply major)is called a seven-step mode, the stable sounds of which form a major (major) triad.

The very word "major" (Italian - maggiore) literally means: “bigger”, “elder”. This term is used in syllabic notation; in alphabetic notation, the word “major” is replaced by the word “dur” (from lat. durus, literally - hard).

The main characteristic feature of the major mode is the interval of the major third between the I and III degrees, which, in fact, determines the specificity (that is, major) of the combined sound of both the stable sounds themselves and the mode as a whole.

Since the stable sounds (I, III and V degrees) form a triad based on the tonic of the mode, all of them taken together are also called tonic triad, and the sounds included in it receive, respectively, the names of the prima, third and fifth of the tonic. For example, in C major.

A minor scale (or simply minor) is a seven-step scale, the stable sounds of which form a small (minor) triad.

The word “minor” itself (Italian - minore) literally means “lesser”. This term is used in syllabic notation, while in alphabetic notation the word “minor” is replaced by the word moll (from the Latin molle, literally “soft”).

The main characteristic feature of the minor mode is the interval of the minor third (m. 3) between the I and III steps, which, in fact, determines the specificity, that is, the minor nature of the combined sound, both of the stable sounds themselves and of the mode as a whole, in any order of execution of its steps .

In principle, the properties and names of the scale degrees in minor will be the same as in major, only - in some cases - the intervallic relationships between them and, accordingly, the nature of their sound change.

The minor scale (like the major scale) has three main types: natural, harmonic and melodic minor.

The minor scale is built as follows: tone-semitone-tone-tone-semitone-tone-tone.

Key

The pitch level of the fret, determined by the sound of the tonic, is called tonality. Placing a fret on the same sounds, but in a different octave, does not have any effect on determining the tonality, since neither the structure of the fret itself, nor the names of its steps and their properties change from this.

The name of any tonality is determined by the name of the sound of the tonic itself (the first degree of the mode), but since the tonality is always inextricably linked with any particular mode (major or minor), an indication of the modal inclination is usually added to its name. Thus, the full name of a tonality, as a rule, contains two components: 1) the name of the tonic and 2) the name of the mode, regardless of which notation system - syllabic or alphabetic - is used: C major (C-dur), A minor (a-moll).

The names of major keys according to the letter system are written with uppercase (capital) letters, and minor keys with lowercase (small) letters. Sometimes, for brevity, the words dur or moll are omitted from the letter system, and then the modal mood is indicated by the spelling of the first letter (uppercase or lowercase).

Parallel and eponymous keys of major and minor

Although historically both main seven-step modes - both major and minor - developed completely independently, without losing their main specific features, there is still a certain kinship between them: the same number of steps, their similar functional meaning, the same directions of mode gravity and etc. The scales of some similar varieties of both modes (for example, harmonic major and harmonic minor, or melodic major and natural minor and, conversely, natural major and melodic minor), built from the same sound, will sound almost the same, differing only in the sound of the third degree - the main and only accurate sign of a particular mode.

Tonality is the pitch of the fret. The name of the tonality comes from the name of the sound taken as the tonic and is made up of the designation of tonic and mode, i.e. words major or minor.

A major mode is a mode, a mode whose stable sounds form a major, or major triad.

There are three types of major mode:

• · Natural major - has the structure T-T-P-T-T-T-P.
• · Harmonic major - major, with a lowered VI degree, has the structure T-T-P-T-P-1/2T-P,
• · Melodic major - degrees VI and VII are lowered; has the structure T-T-P-T-P-T-T.

MINOR MODE

The system of relationships between stable and unstable sounds is called a mode. At the heart of any individual melody and piece of music there is always a certain harmony.

The arrangement of the sounds of a scale in order of height (starting from the tonic of the first degree to the tonic of the next octave) is called a scale. The sounds of the scale are called degrees and are designated by Roman numerals. Of these, stages I, III and V are stable, and stages II, IV, VI and VII are unstable. Unstable steps are resolved by gravity into adjacent stable sound.

A minor mode is a mode, a mode whose stable sounds form a small, or minor triad.

There are three types of minor scale:

• · Natural minor - has the structure T-P-T-T-P-T-T.
• · Harmonic minor - minor, with a raised seventh degree; has the structure T-P-T-T-P-1/2T-P.
• · Melodic minor - the VI and VII degrees rise; has the structure T-P-T-T-T-T-P.